"""Segment tree with lazy propagation: range add, range sum.

Recursive implementation over a power-of-two padded array. The padding is load
bearing: every internal node then spans an even length and splits into two
equal halves, which is exactly what ``_push``'s child-length arithmetic
(``mid - lo + 1`` for the left child, ``hi - mid`` for the right) relies on.

Supports: build, point query (via range sum of length 1), range add, range sum.

Bounds contract: indices are 0-indexed and inclusive. Any endpoint outside
``[0, len(data) - 1]`` raises ``IndexError`` -- nothing is clamped, so a query
can never reach a padded cell that holds no data. An empty tree therefore
accepts no index at all: there is no element to point at. A reversed range
(``l > r``) is empty: it contributes 0 and is not bounds-checked.
"""


class SegmentTree:
    def __init__(self, data):
        self.n = len(data)
        # pad to a power of two: every internal node then spans an even length
        # and splits equally, so _push's two child-length expressions agree.
        self.size = 1
        while self.size < max(1, self.n):
            self.size <<= 1
        self.tree = [0] * (2 * self.size)
        self.lazy = [0] * (2 * self.size)
        # build leaves
        for i in range(self.n):
            self.tree[self.size + i] = data[i]
        for i in range(self.size - 1, 0, -1):
            self.tree[i] = self.tree[2 * i] + self.tree[2 * i + 1]

    def _span_ok(self, l, r):
        """Bounds contract for a [l, r] operation. False means empty range."""
        if l > r:
            return False
        if l < 0 or r >= self.n:
            raise IndexError(
                "range [%d, %d] out of bounds for %d element(s) "
                "(valid indices 0..%d)" % (l, r, self.n, self.n - 1)
            )
        return True

    def _push(self, node, lo, hi):
        if self.lazy[node] != 0:
            mid = (lo + hi) // 2
            left = 2 * node
            right = 2 * node + 1
            # sizes: left covers [lo, mid] = mid - lo + 1, right covers
            # [mid + 1, hi] = hi - mid; equal because spans are even.
            self.tree[left] += self.lazy[node] * (mid - lo + 1)
            self.lazy[left] += self.lazy[node]
            self.tree[right] += self.lazy[node] * (hi - mid)
            self.lazy[right] += self.lazy[node]
            self.lazy[node] = 0

    def _update(self, node, lo, hi, l, r, val):
        if r < lo or hi < l:
            return
        if l <= lo and hi <= r:
            self.tree[node] += val * (hi - lo + 1)
            self.lazy[node] += val
            return
        self._push(node, lo, hi)
        mid = (lo + hi) // 2
        self._update(2 * node, lo, mid, l, r, val)
        self._update(2 * node + 1, mid + 1, hi, l, r, val)
        self.tree[node] = self.tree[2 * node] + self.tree[2 * node + 1]

    def range_add(self, l, r, val):
        """Add val to [l, r] inclusive (0-indexed). Raises IndexError out of bounds."""
        if not self._span_ok(l, r):
            return
        self._update(1, 0, self.size - 1, l, r, val)

    def _query(self, node, lo, hi, l, r):
        if r < lo or hi < l:
            return 0
        if l <= lo and hi <= r:
            return self.tree[node]
        self._push(node, lo, hi)
        mid = (lo + hi) // 2
        return self._query(2 * node, lo, mid, l, r) + self._query(2 * node + 1, mid + 1, hi, l, r)

    def range_sum(self, l, r):
        """Sum of [l, r] inclusive (0-indexed). Raises IndexError out of bounds."""
        if not self._span_ok(l, r):
            return 0
        return self._query(1, 0, self.size - 1, l, r)

    def point_get(self, i):
        return self.range_sum(i, i)

    def to_list(self):
        return [self.point_get(i) for i in range(self.n)]